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Page 6C.1 Chapter 6. Complementary Material
Chapter 6
Complementary Material
6.3 APPC Schemes: Polynomial Approach
Proof of Theorem6.3.2.
Step 1.We establish the expressions (6.48)We rewrite the control law (6.39) and thenormalized estimation error as
2
1 1 ( ),
1 1 .
m p p m
s p p pp p
p p
LQ u P y y
m z y uR Z
=
= =
(6C.1)
( ) ( )p
s s , are monic Hurwitz polynomials of degree 1n q+ , n , respectively,
1 1 ( ) ( ).n
a n b np ps s sZR = + , =
(6C.2)
The proof is simplified if without loss of generality we design
( ) ( ) ( ),p q
s s s = (6C.3)
where ( )q
s is an arbitrary monic Hurwitz polynomial of degree 1q if 2q and
( )q s =1 for q
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Page 6C.2 Chapter 6. Complementary Material
Use the expressions
11 22 2
( ) ( ) ( ) ( ) ( ) ( ),n qp q n q p q n q
R s s s s Z s s s +
+ + = + , = (6C.7)
1 1
0 2 2
( ) ( ) ( ) ( ) ( )n q n qn q m n q
P s p s s L s Q s s sp l + +
+ +
= + , = + (6C.8)
in (6C.5) to obtain
( 1) 21 22 2
( 1) 21 20 2 0 2 0 1
( ) ( ) ,
( ) ( ) ( ) ( ) .
n q
f n q f n q f s
n q
f n q f n q f s m
y s y s u m
u p p s y p l s u p m y
+ + +
+ + +
= + +
= + +
(6C.9)
Define the state
( 2) ( 2) .n q n q ff f f ffx y y u uy u + +
, , , , , , ,
Then (6C.9) can be expressed in the form
21 2 1( ) ( ) ,s mx A t x b t m b y= + + (6C.10)
where
1 2
2 ( 2) ( 1)
1 2
( 2) ( 1) 2
0
0( ) ,
0
0
T T
n q n q x n q
T T T T
o o
n q x n q n q
I
A tp p p l
I
+ + +
+ + +
=
(6C.11)
where ( 2) ( 1)n q x n q+ + is an ( 2) ( 1)n q n q+ + matrix with all elements equal to
zero. Since
( 1)2 ( ) ,
n q
p f f n q fu u u s u
+ + = = +
( 1)2 ( ) ,
n q
p f f n q fy y y s y + + = = +
(6C.12)
where is the coefficient vector of 1( ) n qs s + , it follows that
1 1 1
1 1 1
[0 0 1 0 0] [0 0 ] ,
[1 0 0 0 0] [ 0 0] .
p
n q n q n q
p
n q n q n q
u x x
y x x
+ + +
+ + +
= , , , , , , + , , ,
= , , , , , , + , , ,
(6C.13)
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Page 6C.3 Chapter 6. Complementary Material
Step 2.Establish the e.s. of the homogeneous part of (6C.11)We can show that for
each frozen time t det( ( )) ,
p q m p q qsI A t R LQ P Z A
= + = (6C.14)
where the last equality is obtained by using the Diophantine equation (6.38). Since
q ,A are Hurwitz,A(t) has eigenvalues with negative real parts at each time t; i.e.,A(t)
is stable at each frozen time t.
We now need to show that 2( ) ( )A t L A t L , from the properties p L and
2 ,p L which are guaranteed by the adaptive law (6.37). In a similar way as in the
known parameter case, the matching equation (6.38) may be expressed as
* ,l l l
S a = (6C.15)
where l
S is the Sylvester matrix of .p m
R Q , p
Z ; *l
a is as defined in (6.32); and
[ , ] , [0...0,1, ] .T Tl q q
q
l p l l = =
Using the assumption that the polynomials p m
R Q and p
Z are strongly coprime at each
time t, we conclude that their Sylvester matrix l
S is uniformly nonsingular; i.e.,
0det( )
lS | |> for some 0 0 > . Therefore, p L implies that
1l l
S S L , . From(6C.15) we have
1 * ,l ll
S a = (6C.16)
which implies that l is u.b. On the other hand, because p L and 2p L , it follows
from the definition of the Sylvester matrix that 2
lS L . Noting that
1 1 * ,l l l ll
S S S a =
we have 2
lL , which is implied by
1 l l
LS S
, and 2( )l t LS . Because the vectors
1 2 p l , , , are linear combinations of p l , and all elements in ( )A t are u.b., we have
( ) ( ( ) ( ) ),plA t c t t | |+| |
which implies that 2( )A t L . Using Theorem A.8.6, it follows that the homogeneous partof (6C.10) is e.s.
Step 3. Use the properties of the2
L
norm and BG lemma to establish
boundedness For clarity of presentation let us denote the 2L norm as || || . FromLemma A.5.10 and (6C.10) we have
2 2( )s s
x c m c x t c m c + , | | + (6C.17)
for some 0> . We define the fictitious signal
2 2 21 .f p p
m y u+ + (6C.18)
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Page 6C.4 Chapter 6. Complementary Material
Using (6C.10), (6C.12), (6C.17) andLemma A.5.10 we can establish that ,s f
m m
and
2 22 2 2 ,f s s f
m c c x c m c m m c + + + (6C.19)
i.e.,2 ( ) 2 2 2
0( ) ( ) ( ) .
tt
f s fm c c e m m d
+
Using the property 2sm L , guaranteed by the adaptive law, using the boundedness of
fm , and applying the BG lemma, we can establish that
fm L . The boundedness of
the rest of the signals follows fromf
m L and the properties of the 2L norm that is
used to show thatf
m bounds most of the signals from above.
Step 4.Establish that the tracking error converges to zeroLet us start with the secondequation in (6C.1), i.e.,
2 1 1
.s p pp pp p
m y uR Z =
Filtering each side of the above equation with 1q
mLQ , where
12( ) ( )
n
nL s t s s l
, +
and [1 ]c
l l= , is the coefficient vector of ( )L s t, , we obtain
21 1 1 1 ( ) .m s m p pp p
q q p p
LQ m LQ y uR Z
=
(6C.20)
Using q p = , and applying Lemma A.11.1 (Swapping Lemma 1), we obtain thefollowing:
1 2
1 1 ,m m m m
p p p pp p p p
q p q p
Q Q Q Qy y r u u rR R Z Z
= + , = + (6C.21)
where
1 11 1 1 2 1 1
( ) ( )( ) ( ) ( ) ( ) ,n n
a bc b p c b p
p p
s sr W s W s y r W s W s u
,
(6C.22)
and W 1 1c bW, are as defined inSwapping Lemma 1 withm
q
Q
W = .Because
p pu y L, and 2a b L , , it follows that 1 2 2r r L, . Using (6C.21) in
(6C.20), we obtain
21 2
1 ( ) m m
m s p pp p
q
Q QLQ m L y u r rR Z
= +
(6C.23)
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Page 6C.5 Chapter 6. Complementary Material
which, due to Q 1 1( )m p m m my Q e y Q e= + = , becomes
2
1 1 2
1 ( ) .m m
m s pp p
q
Q QLQ m L e r r L uR Z
= +
(6C.24)
Applying Lemma A.11.4 (Swapping Lemma 4) (i) to the second term in the right-handside of (6C.24), we obtain
3 ( , ) ,m m
p p p p
Q QL Z u Z L s t u r
= + (6C.25)
where
1 3 2 2( ) , ( ) ( , ) ( ) ,( )
T T m
p n b n b n p
QZ s r s F l s u
s = =
and ( , ) ( )b b
F l c l + . Using the identity ( , ) ( ) ( , ). ( )m m
L s t Q s L s t Q s= and the
control law given by the first equation in (6C.1), we obtain
1
1 ( , ) ( , ). .m mp p
Q QL s t u L s t u P e= =
(6C.26)
From (6C.26), (6C.25), (6C.24) we obtain
21 1 1 2 3
1 1( ) ( ) .mm s pp
q
QLQ m L e Z P e L r r rR
= + + (6C.27)
Using Lemma A.11.4(ii) with 01/ , 1f= = , we can obtain the expressions
1 1 4
1 1 5
4 1 1 5 1 1
1 ( , ). ( , ) ,
1 1 ( , ). ( , ) ,
( ) ( , , , ), ( ) ( , , , ),
m
p mp
p p
T T
n a n b
QL e L s t R s t Q e rR
Z P e Z s t P s t e r
r s G s e l r s G s e p
= + = +
(6C.28)
where ,G G are as defined in Lemma A.11.4 and ( , ) ( ) ( )Tn
X s t s t denotes the
swapped polynomial ( , ) ( ) ( )Tn
X s t t s . Due to 1e L and 2, , ,a bl p L it follows
from the expression of ,G G that 2,G G L . Using (6C.28) in (6C.27), we obtain
( )2
1 1 2 3 4 511 ( ) ( , ) ( , ) ( , ) ( , ) ( ) .
m s p m p
q
LQ m L s t R s t Q Z s t P s t e L r r r r r = + + + + (6C.29)
Using the matching equation (6.38) and the fact that * *( ) ( )A s A s= due to the constantcoefficients of * ( )A s , we can write
* * ( , ). ( , ) ( , ). ( , ) ( ) ( ).p m p
L s t R s t Q Z s t P s t A s A s+ = = (6C.30)
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Page 6C.6 Chapter 6. Complementary Material
Substituting (6C.30) in (6C.29), we obtain
* 21 1 2 3 4 5
1 1, ( ) ( )
m s
q
A e LQ m L r r r r r = +
(6C.31)
or
1 * .e A
= (6C.32)
Using the fact that 1( ) ( )T
n cL s s l = , [1, ]
T T
cl l= , we have
21 1 2 3 4 5( ) .
T m
n c s
q
Qs l m r r r r r
= + +
Therefore, (6C.32) may be written as
21 1 0*
0 1 2 1 2 1 1 2*
1 2
2
( ) ,
[ ( ) ( ) ( ) ( ) ],
( , ) ( ) ,
.
T m
n c s
q
T T T
n c n n
m
b n p
Qe s l m w
A
w s l r r s w s wA
Qw F l s u
w G G
= +
= +
=
= +
(6C.33)
Since 1 2 2, ; , , , ,c pl u L F G G r r L ; and * *1 2( ), ( ) mQT
n nA As s are strictly proper and
stable, it follows from Corollary A.5.5 that 0 1 2 2, ,w w w L and 0 0w as t . Letus now apply Lemma A.11.1 (Swapping Lemma 1) to the first equation in (6C.33) toobtain
2 211 0*
( )[ ( ) ] ,T n mc s c b s c
q
Qe l m W s W s m l w
A
= + +
(6C.34)
where ( ), ( )c bW s W s consist of strictly proper stable transfer functions. Since all the
transfer functions in (6C.34) are proper with inputs which are bounded and in 2L , it
follows that 1 2e L L . We can write the plant equation p p p pR y Z u= as
( )p p p p p p py R y Z u + =
or
( ) ( )
, .p p p p
p p p p p p p p
p p
R s R
y y Z u y y sZ u
= + = +
Sincep
is monic, stable with the same degree asp
R , all the transfer functions in the
above equations are proper, and the boundedness of ,p p
y u implies thatp
y is bounded. In
turn, this implies that 1e is bounded, which together with 1 2e L L implies that
1 0e as t .
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Page 6C.7 Chapter 6. Complementary Material
6.4 APPC Schemes : State-Space Approach
Proof of Theorem6.4.3.
Step 1. Develop the closed-loop state error equationsWe express the tracking error
equation
1
1
p
p
p m
Z Qe u
R Q=
in the state-space form
1
1 2 1, ,0
n q
p
Ie e e C eu
+ = + =
(6C.35)
where n qe R + , [1 0 0]T n qC R += , , , , and 1 2
, are the coefficient vectors of
1
n q
p m pR Q s Z Q
+ , , respectively.
Let oe e e be the state-observation error. Using (6.67) and (6C.35), we obtain
1 21
( ) ,
,
T
c o o
o po o
e A t e K C e
A e ee u
= +
= +
(6C.36)
where
1
0
n q
o
IA a
+
is a stable matrix; ( )c c
A t A BK ; and 1 1 1 2 2 2 , . Since the polynomials
1,p p mZ Q R Q have no common unstable zeros it follows using Theorem A.2.5 that there
exist polynomials X(s), Y(s) of degree 1n q+ , with X(s) monic, which satisfy theequation
*1 .p m pR Q X Z Q Y A+ = (6C.37)
( )A s is a Hurwitz polynomial of degree 2( ) 1n q+ and contains all the common stable
zeros of 1,p p mZ Q R Q if any. Dividing each side of (6C.37) with ( )A s and using it to
operate onp
u , we obtain
1
* * .
p m p
p p p
R Q X Q YZu u u
A A+ = (6C.38)
Using 1m p pQ u Q u= and p p p pZ u R y= in (6C.38), we obtain
1 1
* * .
p p
p p p
R XQ Q YRu u y
A A= + (6C.39)
The plant output satisfies
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Page 6C.8 Chapter 6. Complementary Material
.p o m
y C e C e y= + + (6C.40)
Step 2.Establish e.s. for the homogeneous part of (6C.36)For each frozen time t, we
have det( ) det( ) ( )c c c
sI A sI A BK A s = + = (where the last equality is obtained using
(6.63)), which implies that ( )cA t is stable at each frozen time t. If 1 ( , ) ( )
pZ s t Q s and
( , ) ( )p mR s t Q s are strongly coprime, i.e., ( )A B, is strongly controllable at each time t, the
controller gains c
K may be calculated at each time tusing Ackermanns formula [3], i.e.,
1 [0 0 0 1] ( ),c c c
K G A A = , , , ,
where1 [ ]n q
cG B AB A B
+ , , ,
is the controllability matrix of the pair ( )A B, . Because ( )A B, is assumed to be strongly
controllable and A B L, due to p L , we have cK L . Now,
1 1 1 [0 0 0 1] ( ) ( ) .c c c c c c c
dK G G G A A G A A
dt
= , , , , +
Becausep
L and 2p L , it follows that 2|| ( ) ||cK t L , which, in turn, implies that
2|| ( ) ||cA t L . From cA being pointwise stable and 2|| ( ) ||cA t L , we have that ( )cA t is
a u.a.s. matrix by applying Theorem A.8.6. Because 0A is a stable matrix the
homogeneous part of (6C.36) is e.s.
Step 3. Use the properties of the2L
norm and the BG lemma to establish
boundednessApplying Lemmas A.5.9, A.5.10 to (6C.36), (6C.39), (6C.40), we obtain
,
,
,
o
p o o
p p o
e c C e
y c C e c e c c C e c
u c e c y c C e c
+ + +
+ +
(6C.41)
where . denotes the 2L norm for some 0> .
We relate the term To
C e with the estimation error by using (6C.36) to express To
C e
as
11 21( ) ( ).
T Tpo o
C e C sI A e u = (6C.42)
Since ( , )oC A is in the observer canonical form, i.e.,
11 ( )
( ) ,( )
n qT
o
o
sC sI A
A s
+
=
where *0 0( ) det( ),A s sI A= we have
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Page 6C.9 Chapter 6. Complementary Material
1 210
( ),n in
pi io
i o
sC e e u
A
=
=
where 1n n q= + , 1 2[ ] 1 2i i i in i = , , , , = , . We apply Lemma A.11.1 (Swapping
Lemma 1) to each term under the summation in (6C.43) to obtain
( )11 11 1 1 11
( ) ( )( ) ( )
( ) ( )( ) ( )
n i n ip
i i ci bi i
po o
s Q ss se e W s W s e
s Q sA s A s
= +
and
( )12 2 21
( ) ( )( ) ( ) ,
( ) ( )( ) ( )
n i n ip
p p pi i ci bi i
po o
s Q ss sW s W su u u
s Q sA s A s
= +
where 0 1ci bi
W W i n q, , = , , + are transfer matrices defined in Lemma A.11.1 with
1( ) ( )( )
n i
p
s
s Q sW s
= . Therefore, oC e can be expressed as
1
1 21 101 1
1 1 1
1 21 1
1 1
( ) ( )
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( ),
( ) ( ) ( ) ( )( )
n i n inp
pi io ip po
p n q n q
p
p po
s Q s s sC e e r u
s Q s s Q sA s
s Q s s se ru
s Q s s Q sA s
=
+ +
= + = +
(6C.43)
where
( ) ( )11 1 1 20
( ) ( )( ) ( ) ( ) .
( )
np
pci bi bii iio
s Q sr W s W s e W s u
A s
=
From the definition of 1 , we have
( )
1 1 1 1 1 1
1
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ),
n q n q n q
n q n q
m p mp
ap m n mp
s s s
s t Q s s R s Q s sR
s t R s Q s s Q sR
+ + +
+ +
=
= , +
= , =
(6C.44)
wherea a a
is the parameter error. Similarly,
2 1 1 1( ) ( ) ( ),n q b ns s Q s + = (6C.45)
whereb b b
. Using (6C.44) and (6C.45) in (6C.43), we obtain
1
1 1 1 1
1
1
1 1 1
1 1
( ) ( ) ( ) 1 1( ) ( )
( ) ( ) ( )( )
( ) ( ) ( ) ( )( ) ( ) ,
( ) ( ) ( ) ( )( )
p mpa bo n n
p po
p m ma bn p n p
p po
s Q s Q sC e s e s r u
Q s s sA s
s Q s Q s Q ss y s u r
Q s s s Q sA s
= +
= +
(6C.46)
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Page 6C.10 Chapter 6. Complementary Material
where the second equality is obtained using
1 1( ) ( ) ( ) ( ) .pm m p m pQ s e Q s y Q s Q s uu= , =
Using the identities
1 1 1 21 1( ) ( )( ) ( )
n p n p
p p
s u s ys s
= , =
and Lemma A.11.1, we obtain the following equalities:
2 1 2
1 1
( ) ( )( ) ( ) ,
( ) ( ) ( )m m
a a n p cq bq a
p
Q s Q ss y W W s
Q s Q s s
= +
1 1 1
1 1
( ) ( )( ) ( ) ,
( ) ( ) ( )m m
b b n p cq bq b
p
Q s Q ss u W W s
Q s Q s s
= +
where ,cq bqW W are as defined in Swapping Lemma A.11.1 with 1( )
( )( ) m
Q s
Q sW s = . Using theabove equalities in (6C.46), we obtain
2
( ) ( ),
( )
p m
po
o
s Q sC e r
A s
= + (6C.47)
where
( )12 1 1 2( ) ( )
( ) ( ) ( ) ( ) .( )
p
cq bq cq bqb a
o
s Q sr r W s W s W s W s
A s
+ +
The normalized estimation error satisfies the equation
2 ,psm =
which can be used in (6C.47) to obtain
22
( ) ( ).
( )
p m
o s
o
s Q sC e m r
A s
= + (6C.48)
Using the definition of the fictitious normalizing signal 2 2 21 || || || ||f p pm u y+ + and
Lemma A.5.2, we can show thatf s f
m m m L / , / for some 0> . Using these
properties off
m applying Lemma A.5.2 in (6C.48), we obtain
.po s f f C e c m m c m + (6C.49)
From (6C.41) and in the definition off
m , we have
22 2201 ,
T
f p pm u y c C e c= + + +
which together with (6C.49) implies
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222f s f p f
m c m m m c + +
or22 ,
f fm c gm c +
where2 2 2 2
ps
g m +| | and
2
g L due to the properties of the adaptive law. If we now
apply the BG lemma to the above inequality we can show thatf
m L . From
fm L and the properties of the 2L norm we can establish boundedness for the rest of
the signals.
Step 4. Convergence of the tracking error to zeroSince all signals are bounded we can
establish that 2( )dsdt
m L , which, together with 2
2sm L , implies that2( ) ( ) 0st m t
and, therefore, ( ) 0p t as t . From the expressions of 1 2r r, we can conclude,
using Corollary A.5.5, that 2 2r L and 2 ( ) 0r t as t . Using 2
2 2sm r L , and 2 0r as t in (6C.48) and applying Corollary A.5.5, we have that
2oC e L| | and ( ) 0
oC e t| | as t .
Consider the first equation in (6C.36), i.e.,
( ) .c o o
e A t e K C e= +
Since ( )c
A t is u.a.s. and 2oC e L| | , 0
oC e| | as t we can conclude that
0e as t . Since 1 p m oe y y C e C e = + (see (6C.39)), it follows that
1( ) 0e t as t and the proof is complete.
6.7.3 Robust APPC: Polynomial Approach
Proof of Theorem6.7.2.
Step 1.Expressp p
u , y in terms of the estimation error Following exactly the same
steps as in the proof of Theorem 6.3.2 for the ideal case, we can show that the inputp
u
and outputp
y satisfy the same equations as in (6C.10), expressed as
21 2
21 1 2
22 3 4
( ) ( ) ,
,
,
s m
p s m
p s m
x A t x b t m b y
y C x d m d y
u C x d m d y
= + +
= + +
= + +
(6C.50)
where 1 2, ( ), ( ), ,x A t b t b and 1 2; 1 2 3 4i kC i d k , = , , = , , , are as defined in (6C.10) and1( )
mm s
P yy = . The modeling error terms due to ( )m us d , do not appear explicitly in
(6C.50). Due to the unmodeled dynamicss
m , , ( )A t no longer belong to 2L as in the
ideal case, but belong to2
2 0( )mS f + . Because
220
2 2
22
d
m m
+ , we have
22 02 02
( ) .s
s
dm A t S f
m
, , + +
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Step 2. Establish the u.a.s. property of ( )A t As in the ideal case, the APPC law
guarantees that ( ( )) ( )det sI A t A s = for each time twhere ( )A s is Hurwitz. If weapply Theorem A.8.6(ii) to the homogeneous part of (6C.50), we can establish that ( )A t
is u.a.s. which is equivalent to e.s., provided
22 0
0 2 2
1 t T
st
dc f d
T m
+ + + , where 0c is a finite constant. Because2 0s
m > , condition (6C.51) may not be satisfied for small 2 0f , unless 0d , the upperbound for the input disturbance, is zero or sufficiently small. As in the MRAC case, we
can deal with the disturbance in two different ways. One way is to modify 2 1s d
m n= + to2
01s dm n = + + , where 0 is chosen to be large enough so that2 20 0
20 2
d d
mc c
< , say, so
that for
2
0 2( ) 2c f
+ <
condition (6C.51) is always satisfied and ( )A t is e.s. The other way is to keep the same2s
m and establish that when 2s
m grows large over an interval of time 1 1 2[ ]I t t= , , say, the
state-transition matrix ( )t , of ( )A t satisfies 2( )1( ) k t
t k e , t and 1t I, .
This property of ( )A t (when 2s
m grows large) is used in Step 3 to contradict the
hypothesis that 2s
m could grow unbounded and conclude boundedness. We proceed as
follows.
Let us start by assuming that2sm grows unbounded. Because all the elements of the
state xare the outputs of strictly proper transfer functions with the same poles as the
roots of ( )s and inputsp p
u y, and the roots of ( )s are located in Re0
[ ] 2s < / , it
follows from Lemma A.5.2 thats
x
m L . Because sm m Ly , , it follows from (6C.51)
thatp p
s s
y u
m m L, . Because
2 2p p
u y, are bounded from above by 2s
m , it follows from the
equation for 2s
m that 2s
m cannot grow faster than an exponential, i.e.,
1 0( )2 20 0( ) ( ) 0
k t t
s sm t e m t t t
for some 0k> . Because 2 ( )s
m t is assumed to grow
unbounded, we can find a 0 0t > > for any arbitrary constant 2 0t t > such that12
2( ) k
sm t e
> . We have
1 2 01 ( )2 22 0( ) ( ),
k t tk
s se m t e m t
<
which implies that
20 1 2 0ln ( ) ln [ ( )].sm t k t t > +
Because 2 0t t > and 0 2 2( )t t t , , it follows that
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20 0 2 2( ) ( ).sm t t t t > ,
Let2
21 sup arg( ( ) )t st m = = . Then,
21( )sm t = and
21 2( ) [ ),sm t t t t , where
1 2t t , i.e., 2 1t t . Let us now consider the behavior of the homogeneous partof (6C.50), i.e.,
( )Y A t Y = (6C.52)
over the interval 1 1 2[ )I t t, for which2 ( )sm t and 2 1t t , where 0 > is an
arbitrary constant. Because det( ( )) ( )sI A t A s = , i.e., ( )A t is a pointwise stable matrix,the Lyapunov equation
( ) ( ) ( ) ( )A t P t P t A t I+ = (6C.53)
has the solution P(t) = P ( ) 0t > for each 1t I . We consider the Lyapunov function
( ) ( ) ( ) ( );V t Y t P t Y t =
then
( ) .V Y Y Y PY Y Y P t Y Y = + +
Using (6C.53) and the boundedness of P A, , we can establish that || ( ) || || ( ) ||P t c A t .
Because 1 2Y Y V Y Y for some 1 20 < < , it follows that
1 1
2 1( ( )) ;V c A t V
i.e.,1 1
2 1( ( ))
( ) ( )
t
c A s ds
V t e V
0t . For the interval 1 1 2[ )I t t= , , we have2 ( )
sm t and, therefore,
22 02 0( ) ( ) .
t
dA d c f t c
+ + +
Therefore,
0( )
1 2( ) ( ) [ )t
V t e V t t t
, , (6C.54)
and t , provided
22 0
0 2 0 ,d
c f
+ +
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which implies that
0 ( )21 2
1
( ) ( ) [ )tY t e Y t t t
| | | | , ,
which, in turn, implies that the transition matrix ( )t , of (6C.52) satisfies
0 ( )
0 1 2( ) [ ),t
t e t t t
, , , (6C.56)
where 0210 0 2
= , = . Condition (6C.55) can be satisfied by choosing large
enough and by requiring 2 0f , to be smaller than some constant, i.e.,1
220 2 4
( )c f + < ,
say. In the next step we use (6C.56) and continue our argument over the interval 1I in
order to establish boundedness by contradiction.
Step 3. Boundedness using the BG lemma and contradiction We apply Lemma
A.5.10 to (6C.50) for 1 2[ )t t t , . We have
1
1 1
2( ) 21( ) ( ) ,
t t
t t s t t x ce x t c m c
/ , , | |+ +
where1
|| ( ) ||t t, denotes the 2L norm 1 2|| ( ) ||t t , defined over the interval 1[ )t t, , for any
1 00 2 < < < . Becauses
x
m L it follows that
1
1 1
2( ) 21( ) ( ) .
t t
t t s s t t x ce m t c m c
/ , , + +
Because1 1 1 1
2|| || || || || || || ( ) ||pt t pt t t t s t t
y u c x c m c, , , ,, + + , it follows that
1
1 1 1
2( ) 21( ) ( ) .
t t
pt t pt t s s t ty u ce m t c m c / , , ,, + +
Now 2 ( ) 1 ( )s d
m t n t = + and
0 1
1 10 0
2 2( )
1 2 2( ) ( ) .t t
d d pt t pt t n t e n t y u
, ,= + +
Because1 0 12
|| ( ) || || ( ) ||t t t t , , for 0 , it follows that
0 1
1 1
2 2( )2 21 1( ) 1 ( ) 1 ( ) .
t t
s d s pt t pt t m t n t e m t y u t t
, ,= + + + +
Substituting for the bound for1 1
|| || || ||pt t pt t
y u, ,, , we obtain
1
1
2( )2 2 2
1 1( ) ( ) ( ) 0t t
s s s t t m t ce m t c m c t t
, + +
or
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1
1
( )2 2 ( ) 2 2 21( ) ( ) ( ) .
t
t t t
s s s s
t
m t c ce m t c e m m d
+ + (6C.57)
Applying BG lemma 3 (Lemma A.6.3),
2 2 2 2
1 1
1
( )2 2 ( )
1 1( ) (1 ( )) .
t t
s st s
t
c m d c m dt t t s
s s
t
m t c m t e e c e e ds t t + +
Fort, s 1 2[ )t t , we have
22 2 2 0
2 0 ( ) .t
s
s
dc m d c f t s c
+ + +
By choosing large enough so that20
4
dc
< and by requiring
22 0( ) ,4c f
+ <
we have
2
2 1 2 2 12 2 2
1
( ) ( ) ( )2 2 22 1 1( ) (1 ( )) (1 ( )) .
t
t t t s t t
s s s
t
m t c m t e c e ds c m t e c
+ + + +
Since 2 1t t ,2
1( )sm t = , and2
2( )sm t > , we have
222( ) (1 ) .sm t c e c
< + +
Therefore, we can choose large enough so that 2 2( )sm t < , which contradicts the
hypothesis that 2 2( )sm t > . Therefore, sm L , which implies that p px u y L, , .The condition for robust stability is, therefore,
12 2
0 2( ) min2 4
c f
+ < ,
for some finite constant 0c > .
Step 4.Establish bounds for the tracking errorA bound for the tracking error 1e is
obtained by expressing 1e in terms of signals that are guaranteed by the adaptive law to
be of the order of the modeling error in the m.s.s. The tracking error equation has exactly
the same form as in the ideal case and is given by
12 2
1 0
( ) ( ) ( ) ( ),
( ) ( )
n
m n
s
s s Q s s se m v
A s A s
= +
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where 0v is the output of proper stable transfer functions whose inputs are elements of
p multiplied by bounded signals. Because
2
2
20( )
sp s m
m S f
, + and2
2
2 22 0( )
smc d
+ ,
due tos
m L , it follows from Corollary A.5.8 that2 2
1 2 0 0( )e S d f + + and the proofis complete.
6.9 Examples Using the Adaptive Control Toolbox
The PPC and APPC algorithms presented in Chapter 6 can be implemented using a set of
MATLAB functions and the Simulink blockAdaptive Controllerprovided in the
Adaptive Control Toolbox. For implementation of the APPC algorithms, these tools need
to be used together with the PI functions or blocks introduced in Chapter 3. In this
section, we demonstrate the use of the Adaptive Control Toolbox in various PPC andAPPC problems via a number of simulation examples.
6.9.1 PPC: Known Parameters
The Adaptive Control Toolbox provides an easy way of designing and simulating a widerange of PPC schemes. For the polynomial approach, the MATLAB functionppcpoly
can be used to solve (6.31) for a given plant of the form (6.28) and design polynomials* ( )A s and ( )s . Denoting the coefficient vectors of ( ), ( ),Z s R s * ( ), ( ), ( )mA s s Q s as
Z,R,As,Lambda,Qm,respectively,
[Ku,Ky,RETYPE] = ppcpoly(Z,R,As,Lambda,Qm)
returns the coefficient vectors ofu
K andy
K . RETYPEis returned as 0 if the equation
solving process is successful and as 1 if it fails.
Consider the SPM
Tz = (6C.58)
for the plant
1 0
1
1 1 0
( ),
( )
m
m
n n
n
b s b s bZ sy u u
R s s a s a s a
+ + += =
+ + + +
where
[ ]0 1 0
1
,
,( )
1 1,
( ) ( ) ( ) ( )
T
m n
n
p
Tm n
p p p p
b b a a
sz y
s
s su u y y
s s s s
=
=
=
and ( )p
s is a monic Hurwitz design polynomial of order n . This SPM is also used in
parameterization for adaptive PPC, in section 6.9.2.
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The MATLAB function uppcpoly or the Simulink block Adaptive
Controller with appropriate options and parameters can be used to generate the
signals and z of the SPM (6C.58), and the control signal u, as demonstrated in
Examples 6.9.1 and 6.9.2 below.
Example 6.9.1 Consider the plant 21s
y u= . We want to choose u so that the closed-
loop poles are equal to the roots of ( )2
( ) 1A s s = + and y tracks the reference signal
1m
y = . The reference input,m
y , satisfies ( ) 0m m
Q s y = for ( )m
Q s s= . The controlsignal can be generated as
( )( )( )
( ) ( )
yu
m
K sK su u y y
s s= +
, ( ) 0.5s s = + .
( )u
K s and ( )y
K s can be calculated using the following code:
Z = 2;
R = [1 -1];
As = [1 2 1];Lambda = [1 0.5];
Qm = [1 0];
[Ku, Ky, RETYPE] = ppcpoly(Z,R,As,Lambda,Qm)
The result is ( ) 0.5u
K s = , ( ) 1.5 0.5y
K s s= .
Example 6.9.2 In Example 6.9.1, we have calculated the control polynomials as
( ) 0.5uK s = and ( ) 1.5 0.5yK s s= . Now we can generate the linear model and control
signals using the following code (added to the code of Example 6.9.1):
Lambdap = Lambda;
dt = 0.005; % Time increment for simulation
(sec).
t = [0:dt:10]; % Process time (sec).
lt = length(t);
ym = ones(1,lt); % Reference signal.
% State initialization for plant:
[nstate, xp0] = ufilt('init',Zp,Rp,0);
xp(:,1) = xp0;
% State initialization for linear parametric model:
[nstate xl0] = uppcpoly('init',[1 0],Lambda,Lambdap,Qm);
xl(:,1) = xl0;
% Signal initialization:
y(1) = 0;
% Process
for k = 1:lt,u(k) = uppcpoly('control',xl(:,k),[y(k) ym(k)],[Ku
Ky],...Lambda,Lambdap,Qm);
[z(k), phi(:,k)] = uppcpoly('output',xl(:,k),[u(k)
y(k)],[1 0], ...Lambda,Lambdap,Qm);
dxl = uppcpoly('state',xl(:,k),[u(k) y(k) ym(k)],[1 0],
...Lambda,Lambdap,Qm);
xl(:,k+1)=xl(:,k) + dt*dxl;
dxp = ufilt('state',xp(:,k),u(k),Zp,Rp);
xp(:,k+1)=xp(:,k) + dt*dxp;
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y(k+1) = ufilt('output',xp(:,k+1),Zp,Rp);
end
% Outputs:
y = y(1:lt);
The results are shown in Figures 6C.1 and 6C.2. Another way of simulating the above
scheme is to use the Simulink blockAdaptive Controller. We can use the
Simulink scheme shown in Figure 6C.3, and choose the appropriate options and enter the
appropriate parameters in the menus of theAdaptive Controller block.
Figure 6C.1 Output and control signals of Example6.9.2.
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Page 6C.19 Chapter 6. Complementary Material
Figure 6C.2 SPM signals of Example6.9.2.
u(t)
z
z(t)
y_m
ym(t)
u
y
y(t)
t
time
[2 -1]
theta
phi
phi(t)
T.1
2
s-1
Plant
1
Constant
Clock
AdaptiveController
Adaptive Controller
Figure 6C.3 Simulink implementation of the PPC Scheme in Example6.9.2.
For the PPC scheme based on the state-feedback approach, where the state vector e
of the state-space representation (6.52) for the plant (6.50), i.e.,
1( ) ( ) ( ), ,T
pe t Ae t Bu t e C e= + =
is available for measurement, the MATLAB functionppcssv can be used to compute
the state-feedback gainc
K of the PPC law
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Page 6C.20 Chapter 6. Complementary Material
cu K e=
for a given design polynomial* ( )A s . Denoting the coefficient vectors of
*( ), ( ), ( )Z s R s A s by Z, R, As, respectively,
[Kc,RETYPE] = ppcssv(Z, R, As)
returns the feedback gainc
K . RETYPEis returned as 0 if the equation solving process is
successful and as 1 if it fails.If the state vector of (6.52) is not available for measurement, then the value of the
Luenberger observer gaino
K in (6.55), which is required in estimation of the state vector
e , can be computed using the MATLAB function stateest. Denoting the desired
characteristic polynomial ofT
c o cA K C by * ( )
oA s , and the coefficient vector of * ( )
oA s by
Aos,
[Ko,RETYPE] = stateest(C, R, Aos)
returns the observer gaino
K . RETYPEis returned as 0 if the equation solving process is
successful and as 1 if it fails.
The MATLAB function uppcrsf or the Simulink block Adaptive
Controller with appropriate options and parameters can be used to generate the
signals and z of the SPM (6C.58), and the control signal u, as demonstrated in
Example 6.9.4.
Example 6.9.3 Consider the plant 21s
y u= . We like to choose u so that the closed-
loop poles are equal to the roots of 2( ) ( 1)A s s = + and y tracks the reference signal1
my = . This problem can be converted to a regulation problem by considering thetracking error equation
12( 1)( 1)
mse y y u
s s+= =
,
1su u
s=
+,
where we want 1e to converge to 0. Assuming that the states of the canonical
representation
1 1 2
0 0 2e e u
= +
, [ ]1 0 e=
are available for measurement, we can design a state-variable controller
1
, .
s
u u u Kes
+= =
The value of K can be calculated as follows:
R = [1 -1 0];
Z = [2 2];
As = [1 2 1];
[K, RETYPE] = ppcssv(Z, R, As)
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The result is [ 1 0.5]K= .If the state vector eis not available for measurement, we use a state estimator. Let us
design a state estimator with gain vector L so that [ ]1 10 0
1 0L
has the characteristic
polynomial * 2( 2)o
A s= + . The following code can be used for this purpose:
R = [1 -1 0];C = [1 0];
Aos = [1 4 4];
[L, RETYPE] = stateest(C, R, Aos)
The calculated observer gain is [ ]5 4 T
L= .
Example 6.9.4 In Example 6.9.3, we have calculated the feedback gain as
[ ]1 0.5K= . Now we can generate the linear model and control signals using thefollowing code:
Qm = [1 0];
Z = 2;R = [1 -1];
Q1 = [1 1];
As = [1 2 1];
ba = [Z R(2)];
[num den] = uppcrsf('augment',[1 0],Q1,Qm,ba);
[K, RETYPE] = ppcssv(num,den,As)
Lambdap = Q1;
Aos = [1 4 4];
dt = 0.005; % Time increment for simulation (sec).
t = [0:dt:10]; % Process time (sec).
lt = length(t);
ym = ones(1,lt); % Reference signal.
% State initialization for plant:
[nstate, xp0] = ufilt('init',Z,R,0);
xp(:,1) = xp0;
% State initialization for state space and linear parametric model:
[nstate xl0] = uppcrsf('init',[1 0],Aos,Q1,Qm,Lambdap);
xl(:,1) = xl0;
% Signal initialization:
y(1) = 0;
% Process
for k = 1:lt,
[u(k) ubar(k)] =
uppcrsf('control',xl(:,k),K,Q1,Qm,Lambdap);
[z(k), phi(:,k)] = uppcrsf('output',xl(:,k),[u(k)y(k)],[1 0],Aos,Q1,Qm,Lambdap);
dxl = uppcrsf('state',xl(:,k),[u(k) ubar(k) y(k)
ym(k)],[1 0],
Aos,Q1,Qm,Lambdap, ba);
xl(:,k+1)=xl(:,k) + dt*dxl;
dxp = ufilt('state',xp(:,k),u(k),Z,R);
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Page 6C.22 Chapter 6. Complementary Material
xp(:,k+1)=xp(:,k) + dt*dxp;
y(k+1) = ufilt('output',xp(:,k+1),Z,R);
end
% Outputs:
y = y(1:lt);
The results are plotted in Figures 6C.4 and 6C.5. Another way of simulating the above
schemes is to use the Simulink blockAdaptive Controller. One can use a scheme
similar to the one shown in Figure 6C.3, and choose the appropriate options and enter the
appropriate parameters in the menus of theAdaptive Controller Simulink
block.
Figure 6C.4 Output and control signals of Example6.9.4.
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Page 6C.23 Chapter 6. Complementary Material
Figure 6C.5 Linear parametric model signals of Example6.9.4.
For the linear quadratic (LQ) control, the MATLAB functionppcclq can be used
to compute the state-feedback gain K for the plant (6.50) with the state-space
representation (6.71) and a given penalty coefficient . Denoting and the coefficient
vectors of ( ), ( )Z s R s by lambda and Z, R,respectively,
[K,RETYPE] = ppcclq(Z, R, lambda)
solves the Riccati equation (6.73) and returns the feedback gain K. RETYPEis returnedas 0 if the equation solving process is successful and as 1 if it fails.
The MATLAB function uppcrsf or the Simulink block Adaptive
Controller with appropriate options and parameters can be used to generate the
signals and z of the SPM (6C.58), and the control signal u,as in the case of PPC,
using a state-feedback approach.
Example 6.9.5 Consider the plant 21s
y u= of Example 6.9.3. We want to choose u so
that the closed-loop plant is stable and y tracks the reference signal 1m
y = . As before,this problem can be converted to a regulation problem by considering the tracking errorequation
1 2( 1)( 1)
mse y y u
s s+= =
,
1su u
s=
+,
where we want 1e to converge to 0. Assuming that the states of the canonical
representation (or the states successfully estimated; see Example 6.9.3)
1 1 2
0 0 2e e u
= +
, [ ]1 0 e= ,
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Page 6C.24 Chapter 6. Complementary Material
are available for measurement, we can design the LQ controller
1, ,
su u u Ke
s
+= =
which minimizes the cost function2 2
0( 0.2 )J e u dt
= +. The corresponding feedback
gainK can be calculated as follows:
R = [1 -1 0];
Z = [2 2];
[K, RETYPE] = ppcclq(Z, R, 0.2)
The result is [ 2.7361 0.5]K= . If we apply this gain in Example 6.9.4 instead of[ 1 0.5] , we obtain the results shown in Figures 6C.6 and 6C.7.
Figure 6C.6 Output and control signals of Example6.9.5.
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Page 6C.25 Chapter 6. Complementary Material
Figure 6C.7 Linear parametric model signals of Example6.9.5.
6.9.2 APPC
An APPC scheme, in general, is composed of a PPC law that can be obtained using one
of the three approaches mentioned in section 6.9.1 and a PI algorithm to estimate theunknown plant parameters, which can be chosen from a wide class of algorithms
presented in Chapter 3.
The Adaptive Control Toolbox functionsppcpoly,ppcssv,ppcclq can be
used to generate the control signals, as demonstrated in section 6.9.1, with unknown plant
parameters. The functionsuppcpolyanduppcrsfor the Simulink blockAdaptive
Controller with appropriate options and parameters incorporated with parameter
identification functions (or the Parameter Estimator block) can be used to
generate the signals and z of the linear model (6C.58), and the control signal u.
Example 6.9.6 Consider the plant of Example 6.9.2. Assume that the plant parameters
are unknown. We want to apply the same control algorithm. We can use the linear model*T
z = , where
[ ]
2
2
*
2 2 2 2
,( 1)
2 2 1 0 ,
1,
( 1) ( 1) ( 1) ( 1)
T
p
T
sz y
s
s s su u y y
s s s s
=
+=
= + + + +
to estimate the unknown plant parameters. The Simulink blocks Parameter
EstimatorandAdaptive Controllercan be used for simulation. Selecting the
PI algorithm to be LS with forgetting factor 1= and 0 10 IP = , and initializing
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Page 6C.26 Chapter 6. Complementary Material
as[1 1 1 1]T , the results plotted in Figures 6C.8 and 6C.9 demonstrate that both the
estimation and tracking objectives are successful.
Figure 6C.8 Output and control signals of Example6.9.6.
Figure 6C.9 SPM parameter estimate 1 0 [ ]Tb a= of Example6.9.6.
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Page 6C.27 Chapter 6. Complementary Material
Bibliography
[1] T. KAILATH,Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980.
[2] P. A. IOANNOU AND J. SUN, Robust Adaptive Control, Prentice-Hall, Englewood
Cliffs, NJ, 1996; also available at
http://www-rcf.usc.edu/ ~ioannou/Robust_Adaptive_Control.htm.
[3] B. D. O. ANDERSON, Exponential stability of linear equations arising in adaptive
identification, IEEE Trans. Automat. Control, 22 (1977), pp. 8388.